Polynomial factorization algorithms over number fields

Roblot, Xavier-François

doi:10.1016/j.jsc.2004.05.002

Cited by 7 publications

(10 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Decompose f (t) mod P by calling any appropriate polynomial factorization algorithm modulo the prime ideal(e. g., those in [20,23,24]),i. e., to compute distinct monic irreducible polynomials h1(t), . .…”

Section: Solving Subproblems In Stepmentioning

confidence: 99%

“…Remark on Complexity of the Polynomial Factorization Algorithm for (5.1): (5.1) can be solved via lots of algorithms, for example, the algorithm in [24] is a good solver which time-complexity is polynomial in the degree n and the number of basic arithmetic operations in the (finite) field O K /P . It's also an elegant random algorithm which successful probability is at least 4/9.…”

Section: Solving Subproblems In Stepmentioning

confidence: 99%

See 1 more Smart Citation

A Local-Global Approach to Solving Ideal Lattice Problems

Tian

Sun

Zhu

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We construct an innovative SVP(CVP) solver for ideal lattices in case of any relative extension of number fields L/K of degree n where L is real(contained in R). The solver, by exploiting the relationships between the so-called local and global number fields, reduces solving SVP(CVP) of the input ideal A in field L to solving a set of (at most n) SVP(CVP) of the ideals Ai in field Li with relative degree 1 ≤ ni < n and i ni = n. The solver's space-complexity is polynomial and its time-complexity's explicit dependence on the dimension (relative extension degree n) is also polynomial. More precisely, our solver's time-complexity is poly(n, |S|, NP G, NP T , N d , N l ) where |S| is bit-size of the input data and NP G, NP T , N d , N l are the number of calls to some oracles for relatively simpler problems (some of which are decisional). This feature implies that if such oracles can be implemented by efficient algorithms (with time-complexity polynomial in n), which is indeed possible in some situations, our solver will perform in this case with timecomplexity polynomial in n. Even if there is no efficient implementations for these oracles, this solver's time-complexity may still be significantly lower than those for general lattices, because these oracles may be implemented by algorithms with sub-exponential time-complexity

show abstract

Section: Solving Subproblems In Stepmentioning

confidence: 99%

Section: Solving Subproblems In Stepmentioning

confidence: 99%

A Local-Global Approach to Solving Ideal Lattice Problems

Tian

Sun

Zhu

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…We shall describe an application of van Hoeij's ideas to the direct factorization in K[X], which is in general superior to the norm approach since the number of modular factors over K is smaller than over Q, possibly by a factor of d, without increasing too much the cost of the other steps. We follow roughly the approach of Roblot [26], himself following Lenstra [19], with various improvements. (Namely, using an arbitrary order instead of O K , faster reconstruction through better bounds and reduction heuristics, and van Hoeij's knapsack.…”

Section: Factorization In K[x]mentioning

confidence: 99%

A relative van Hoeij algorithm over number fields

Belabas

2004

Journal of Symbolic Computation

View full text Add to dashboard Cite

Van Hoeij's algorithm for factoring univariate polynomials over the rational integers rests on the same principle as Berlekamp-Zassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over number fields.

show abstract

“…Then, by the resultant computations, a new univariate polynomial h which has a root y j for some j ∈ {1, 2} is obtained. Then, we exploit a polynomial factoring algorithm over number fields [16,1] to recover y j , which takes polynomial time in a degree of h.…”

Section: Introductionmentioning

confidence: 99%

Finding small roots for bivariate polynomials over the ring of integers

Kim

Lee

2024

AMC

View full text Add to dashboard Cite

In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{I} $\end{document}</tex-math></inline-formula> over the ring of integers <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R} $\end{document}</tex-math></inline-formula>. Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields.Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation.As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{I} $\end{document}</tex-math></inline-formula> in polynomial time under some constraint.

show abstract

Polynomial factorization algorithms over number fields

Cited by 7 publications

References 14 publications

A Local-Global Approach to Solving Ideal Lattice Problems

A Local-Global Approach to Solving Ideal Lattice Problems

A relative van Hoeij algorithm over number fields

Finding small roots for bivariate polynomials over the ring of integers

Contact Info

Product

Resources

About